Difficulty: Intermediate | Prerequisites: Ray model of light, reflection (18.1, 18.2), refraction and Snell's law (18.3). You should be comfortable with measuring angles from the normal and applying Snell's law before starting this section.
This is the payoff section of Chapter 18, where reflection and refraction come together to explain how lenses and curved mirrors form images. The thin-lens equation is the single most important formula in this unit and ties object distance, image distance, and focal length into one relation. Ray tracing gives you a visual check on the algebra. If you can draw three special rays and solve the thin-lens equation, you can handle nearly any optics problem at this level.
Converging lenses and concave mirrors focus parallel rays to a real focal point; diverging lenses and convex mirrors spread them apart from a virtual focal point. The thin-lens equation 1/s + 1/s' = 1/f, combined with sign conventions, tells you where the image forms and whether it is real or virtual, upright or inverted. Three special rays let you trace the image graphically as a sanity check.
Lens
A transparent object that uses refraction at curved surfaces to form an image. Light refracts at the air-to-glass boundary and again at the glass-to-air boundary.
Converging lens
A lens that is thicker in the centre than at the edges. It refracts parallel rays toward the optical axis so they meet at the far focal point. Think of it as a lens that "pulls rays together."
Diverging lens
A lens that is thinner in the centre than at the edges. It refracts parallel rays away from the optical axis so they appear to come from the near focal point. Think of it as a lens that "pushes rays apart."
Focal point
The point where parallel rays converge (converging lens) or appear to diverge from (diverging lens) after passing through the lens. Every lens has two focal points, one on each side, equidistant from the lens.
Focal length (f)
The distance from the lens (or mirror) to its focal point. Determined by the curvature and the index of refraction of the lens material.
Thin lens
An idealised lens whose thickness is zero. All refraction is treated as happening at a single plane (the lens plane), and all distances are measured from that plane.
Real image
An image formed where refracted (or reflected) rays actually converge. You can project a real image onto a screen. The image distance s' is positive for a real image.
Virtual image
An image formed where rays appear to diverge from but do not actually pass through. You cannot project it onto a screen. The image distance s' is negative for a virtual image.
Magnification (m)
The ratio of image height to object height: m = -s'/s. A positive m means the image is upright; a negative m means the image is inverted. |m| > 1 means the image is larger than the object.
Concave mirror (converging mirror)
A curved mirror whose reflecting surface curves inward, toward the light source. It has a positive focal length and can produce either real or virtual images.
Convex mirror (diverging mirror)
A curved mirror whose reflecting surface curves outward, away from the light source. It has a negative focal length and always produces virtual, upright, reduced images.
A converging lens is thicker at the centre. Parallel rays refract toward the optical axis and converge at the far focal point.
A diverging lens is thinner at the centre. Parallel rays refract away from the axis and appear to diverge from the near focal point.
Both types have two focal points, one on each side of the lens, each a distance f from the lens plane.
The focal length is a property of the lens itself, determined by its curvature and index of refraction.
All refraction is treated as occurring at the lens plane. The lens has zero thickness, and all distances (object distance s, image distance s', focal length f) are measured from this plane.
A ray parallel to the optical axis passes through the far focal point after the lens.
A ray through the near focal point exits the lens parallel to the optical axis.
A ray through the centre of the lens is not deflected.
Where any two of these rays intersect on the far side of the lens, you have a real image.
Object outside the focal length (s > f): the three special rays converge on the far side. The image is real and inverted.
Object inside the focal length (s < f): the rays diverge after the lens. Extending them backwards, they appear to come from a point on the same side as the object. The image is virtual and upright.
The reflecting surface curves toward the light source (edges curve inward).
Three special rays, analogous to the lens rules:
A ray parallel to the optical axis reflects through the focal point.
A ray through the focal point reflects parallel to the optical axis.
A ray directed at the centre of the mirror reflects at an equal angle on the opposite side of the axis.
When the object is beyond the focal point (s > f): the image is real, inverted, and on the same side as the object.
The reflecting surface curves away from the light source (edges curve outward).
The same three ray rules apply, but the focal point is behind the mirror.
The image is always virtual, upright, and smaller than the object. Diverging rays appear to come from a point behind the mirror.
Concave mirror = converging mirror, positive f, can make real or virtual images.
Convex mirror = diverging mirror, negative f, always makes virtual upright images.
The thin-lens equation relates the three fundamental quantities of any single-lens or single-mirror system:
1/s + 1/s' = 1/f
where s is the object distance, s' is the image distance, and f is the focal length.
Object distance (s): always positive for a single lens or mirror.
Image distance (s'): positive for real images (on the opposite side of the lens from the object, or on the same side as the object for a mirror), negative for virtual images.
Focal length (f): positive for converging lenses and concave mirrors, negative for diverging lenses and convex mirrors.
The magnification is m = -s'/s.
m > 0: image is upright.
m < 0: image is inverted.
|m| > 1: image is larger than the object.
|m| < 1: image is smaller than the object.
Converging lens/concave mirror, object outside f: image is real, inverted (m < 0, s' > 0).
Converging lens/concave mirror, object inside f: image is virtual, upright (m > 0, s' < 0).
Diverging lens/convex mirror, any object position: image is always virtual, upright, and reduced (m > 0, |m| < 1, s' < 0).
A stamp collector uses a converging lens 2.0 cm above a stamp. The magnification is 4.
m = +4 (upright image, so virtual). m = -s'/s, so s' = -4 × 2.0 = -8.0 cm.
1/f = 1/s + 1/s' = 1/2.0 + 1/(-8.0) = 0.5 - 0.125 = 0.375 cm⁻¹.
f = 1/0.375 = 2.7 cm.
An object is 20 cm to the left of a diverging lens. The image is 12 cm from the lens on the same side as the object (so s' = -12 cm).
1/f = 1/20 + 1/(-12) = 0.05 - 0.0833 = -0.0333 cm⁻¹.
f = -30 cm (negative, as expected for a diverging lens).
Thin-lens (and mirror) equation:
\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}Magnification:
m = -\frac{s'}{s}Sign convention summary:
Quantity | Positive | Negative |
|---|---|---|
s (object distance) | Always (single lens/mirror) | N/A |
s' (image distance) | Real image | Virtual image |
f (focal length) | Converging lens / concave mirror | Diverging lens / convex mirror |
m (magnification) | Upright image | Inverted image |
Magnifying glasses are converging lenses used with the object inside the focal length, producing a virtual, upright, enlarged image. Camera lenses and projectors use converging lenses with the object outside the focal length to create real, inverted images on film or a screen. Convex mirrors are used as car side mirrors ("objects in mirror are closer than they appear") because they always produce upright, reduced images with a wide field of view.
Students often forget the sign conventions and plug in all positive numbers. The sign of s' and f carries essential information about whether the image is real or virtual and whether the lens is converging or diverging.
A common error is thinking that a diverging lens can produce a real image. It cannot. Diverging lenses and convex mirrors always produce virtual, upright images.
Students sometimes confuse "inverted" with "reversed left to right." An inverted image is flipped top to bottom (and left to right), not just mirrored.
When using the thin-lens equation, students occasionally solve for s' and get a negative number, then panic. A negative s' simply means the image is virtual, which is the correct physical result for certain configurations.
⚠️ The thin-lens equation (1/s + 1/s' = 1/f) and the magnification formula (m = -s'/s) appear on nearly every optics exam. Be fluent with both.
⚠️ You will almost certainly be asked to determine whether an image is real or virtual, upright or inverted, and larger or smaller, for a given lens or mirror setup. The sign conventions are your guide.
⚠️ Ray-tracing diagrams are commonly tested. Know the three special rays for both converging and diverging lenses, and for concave and convex mirrors.
⚠️ The transition point, where the object is exactly at the focal length, produces no image (rays emerge parallel). Examiners like to test this edge case.
True or false: a diverging lens has a positive focal length. (False: it is always negative.)
Fill in the blank: for a converging lens with the object beyond f, the image is ______ and ______. (real, inverted)
True or false: m = +3 means the image is inverted and three times larger. (False: positive m means the image is upright.)
Fill in the blank: a convex mirror always produces ______, ______, ______ images. (virtual, upright, reduced)
True or false: placing the object at the focal point of a converging lens produces no image. (True: the refracted rays emerge parallel and never converge.)
Q: An object is placed 15 cm from a converging lens with f = 10 cm. Where is the image, and is it real or virtual?
A: 1/s' = 1/f - 1/s = 1/10 - 1/15 = (3 - 2)/30 = 1/30. So s' = 30 cm. Since s' is positive, the image is real. m = -30/15 = -2, so the image is inverted and twice the size of the object.
Q: An object is 5 cm from a converging lens with f = 10 cm. Describe the image.
A: 1/s' = 1/10 - 1/5 = (1 - 2)/10 = -1/10. So s' = -10 cm. Negative s' means the image is virtual. m = -(-10)/5 = +2, so the image is upright and twice as large. This is the magnifying-glass configuration.
Q: A convex mirror has f = -20 cm. An object is 30 cm from the mirror. Find the image distance and magnification.
A: 1/s' = 1/f - 1/s = 1/(-20) - 1/30 = -0.05 - 0.0333 = -0.0833. s' = -12 cm. The image is virtual (behind the mirror). m = -(-12)/30 = +0.4, so the image is upright and reduced to 40% of the object's size.
Q: Explain why a diverging lens cannot form a real image on its own.
A: A diverging lens refracts all rays away from the axis. After passing through the lens, the rays diverge and never converge to a point on the far side. The only "image" is the virtual point from which the diverging rays appear to originate, on the same side as the object.
The thin-lens equation is the algebraic counterpart to the ray-tracing diagrams from Section 18.5; always use one to check the other. Concave and convex mirrors (18.6) obey the same equation with the same sign conventions, so learning it once covers both lenses and mirrors. The magnification formula connects to the human eye and corrective lenses (covered later), where the goal is to move the image to a comfortable viewing distance.
Thin lens, thin-lens equation, converging lens, diverging lens, concave mirror, convex mirror, focal length, focal point, real image, virtual image, inverted image, upright image, magnification, ray tracing, ray diagram, three special rays, sign conventions optics, image distance, object distance, 1/s + 1/s' = 1/f, PHYS 116, University of Washington, general physics optics, Chapter 18